390 Quick
Answers 18 April
GREAT
Day is on Wednesday. Read the program.
Make GREAT Day plans. Our talks are in session 3D, but
only those presenting have obligations to us. Learn
something interesting - doesn't need to be mathematical (or
historical). Tell us about it on Friday. I will say the same on Monday.
I currently have 3 papers that I have not read
(two of those have feedback for talks). It may be another
week for those remaining. I did also update the actual
current average for the drafts I have now read. That
reflects both the recent reactions and the draft.
(Catch: no one's actual current average reflects today's
reactions, so for those of you who skipped today, that will have
an impact.) If you have draft comments back you have a new
actual current average. Please read your
paper feedback. Many need to deeply change their papers to
focus on mathematics. Your draft assignment for 7 April was
to write the paper as well as you could without my feedback.
Your assignment for 7 May is to improve the paper based on my
feedback. Therefore, if you do _not_ improve the paper based
on my feedback, in my understanding, you have not done the
assignment for 7 May (even if you turn in something, e.g. the same
paper as before). So, if you do not improve the paper based
on my feedback, you will earn no credit for the final paper
assignment.
Any more final topics?
Here's something to keep in mind for examples on the exam:
starting with really chapter 5, culture and region is something
that loosely corresponds to modern countries. So, Germany
and France are different. You really never get to say
"they were from Europe" and have that be enough.
Lecture
Reactions
I
regretted not saying this about Abel: he proved that there
exists a quintic polynomial that cannot be factored by
radicals. Curiously this work was so general that he did not
exhibit an example. FYI, more from Galois work, most quintic
polynomials, can't be factored by radicals, although certainly not
all of them. (x-1)^5 obviously can. x^5-10x+ 2 and
x^5+2x+2 cannot. They don't look particularly fancy.
Yes, manifolds continue into higher dimensions, not just dimension
two where they are easier to draw and visualise.
In the Riemann sphere there is only one point at infinity.
In Calc I, 1/0 = ± infinity, but because there is only one, this
is much simplified.
I
think, despite so many contributions, that Riemann worked almost
exclusively on the analytic side of mathematics. I don’t
know of contributions on the side of algebra. Mathematics is
getting big.
I
will talk about the history of linear algebra next time as a
bonus. There are a few extra stories like this that I enjoy
sticking in when we have time as we near the end.
Reading
Reactions
I'm
reminded of this because of Banneker, but I think it's worth
pointing out … ancient Egyptians were black. And our Islamic
mathematicians were diverse peoples.
I do not believe Jefferson's comments to Condorcet about Banneker
reveal his true thoughts. The point is that you are right to
be suspicious.
It’s hard to do mathematics when you’re more concerned with
survival.
Before
the revolution education was mostly handled overseas, but one way
to be independent is to be self-educating.
Before
the civil war it was common to say “the united states
are”. We don’t think about it much, but “united states”
really meant that this was a collection of independent countries
like the EU. This is why we live with such a strange
system of government now where senators have so much power,
despite us not thinking of this place as 50 countries.
But, yes, the colonies all had their own currency.
I like the idea of thinking of periodical journals as a
precursor of modern online discussions. It's a way to
share problems and work on solutions. Establishing this
consistently in the US was tricky, mostly because people didn't
reliably have time to focus on such things.
I hope you studied zero-divisors in algebra (330).
When working mod 6, 2*3=0, so they are zero divisors.
A
nilpotent (zero-power) is something that when raised to some power
yields zero. An idempotent (identity-power) is something
that when raised to some power yields itself. There are nice
examples with matrices.